Looking at your desk further, looks like the desk legs are sitting on 4 casters. Mark the position of these casters on the stand, drill 4 holes about 1/4-1/2 deep in the stand, and then the casters will fit snugly and securely into the stand.

"several people (for some reason, all women) have mentioned that they were concerned about the desk or stand falling over"

Well, I'm a guy, and I think your stand as-is should not be used. It's not a question of if the desk will fall, but when. I'd favor putting 1x1 rails on the top of the stand so the desk can not slide back and forth, but the desk can still be easily removed. In earthquake country, the desk should be attached to the stand, and the whole assembly fastened to the wall. After all, you have several hundred dollars of equipment setting on the desktop. Why risk a tumble when another half-hour or less work will secure this.

BTW, new research has just been released stating how bad sitting all day is, even for those who spend hours in the gym. I think I'll build one of these.